Gil Goffer, Be'eri Greenfeld, Alexander Yu. Olshanskii
{"title":"Asymptotic Burnside laws","authors":"Gil Goffer, Be'eri Greenfeld, Alexander Yu. Olshanskii","doi":"arxiv-2409.09630","DOIUrl":null,"url":null,"abstract":"We construct novel examples of finitely generated groups that exhibit\nseemingly-contradicting probabilistic behaviors with respect to Burnside laws.\nWe construct a finitely generated group that satisfies a Burnside law, namely a\nlaw of the form $x^n=1$, with limit probability 1 with respect to uniform\nmeasures on balls in its Cayley graph and under every lazy non-degenerate\nrandom walk, while containing a free subgroup. We show that the limit\nprobability of satisfying a Burnside law is highly sensitive to the choice of\ngenerating set, by providing a group for which this probability is $0$ for one\ngenerating set and $1$ for another. Furthermore, we construct groups that\nsatisfy Burnside laws of two co-prime exponents with probability 1. Finally, we\npresent a finitely generated group for which every real number in the interval\n$[0,1]$ appears as a partial limit of the probability sequence of Burnside law\nsatisfaction, both for uniform measures on Cayley balls and for random walks. Our results resolve several open questions posed by Amir, Blachar,\nGerasimova, and Kozma. The techniques employed in this work draw upon geometric\nanalysis of relations in groups, information-theoretic coding theory on groups,\nand combinatorial and probabilistic methods.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"18 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09630","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We construct novel examples of finitely generated groups that exhibit
seemingly-contradicting probabilistic behaviors with respect to Burnside laws.
We construct a finitely generated group that satisfies a Burnside law, namely a
law of the form $x^n=1$, with limit probability 1 with respect to uniform
measures on balls in its Cayley graph and under every lazy non-degenerate
random walk, while containing a free subgroup. We show that the limit
probability of satisfying a Burnside law is highly sensitive to the choice of
generating set, by providing a group for which this probability is $0$ for one
generating set and $1$ for another. Furthermore, we construct groups that
satisfy Burnside laws of two co-prime exponents with probability 1. Finally, we
present a finitely generated group for which every real number in the interval
$[0,1]$ appears as a partial limit of the probability sequence of Burnside law
satisfaction, both for uniform measures on Cayley balls and for random walks. Our results resolve several open questions posed by Amir, Blachar,
Gerasimova, and Kozma. The techniques employed in this work draw upon geometric
analysis of relations in groups, information-theoretic coding theory on groups,
and combinatorial and probabilistic methods.